Complex number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted, called the imaginary unit and satisfying the equation ; because no real number satisfies the above equation, was called an imaginary number by René Descartes. Every complex number can be expressed in the form, where and are real numbers, is called the ', and is called the '. The set of complex numbers is denoted by either of the symbols or. Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world.
Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation
has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions and.
Addition, subtraction and multiplication of complex numbers can be naturally defined by using the rule along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse. This makes the complex numbers a field with the real numbers as a subfield. Because of these properties,, and which form is written depends upon convention and style considerations.
The complex numbers also form a real vector space of dimension two, with as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin. The operation of complex conjugation is the reflection symmetry with respect to the real axis.
The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.
Definition and basic operations
A complex number is an expression of the form, where and are real numbers, and is an abstract symbol, the so-called imaginary unit, whose meaning will be explained further below. For example, is a complex number.For a complex number, the real number is called its real part, and the real number is its imaginary part. The real part of a complex number is denoted,, or ; the imaginary part is,, or : for example,,.
A complex number can be identified with the ordered pair of real numbers, which may be interpreted as coordinates of a point in a Euclidean plane with standard coordinates, which is then called the complex plane or Argand diagram. The horizontal axis is generally used to display the real part, with increasing values to the right, and the imaginary part marks the vertical axis, with increasing values upwards.
A real number can be regarded as a complex number, whose imaginary part is 0. A purely imaginary number is a complex number, whose real part is zero. It is common to write,, and ; for example,.
The set of all complex numbers is denoted by or .
In some disciplines such as electromagnetism and electrical engineering, is used instead of, as frequently represents electric current, and complex numbers are written as or.
Addition and subtraction
Two complex numbers and are added by separately adding their real and imaginary parts. That is to say:Similarly, subtraction can be performed as
The addition can be geometrically visualized as follows: the sum of two complex numbers and, interpreted as points in the complex plane, is the point obtained by building a parallelogram from the three vertices, and the points of the arrows labeled and . Equivalently, calling these points,, respectively and the fourth point of the parallelogram the triangles and are congruent.
Multiplication
The product of two complex numbers is computed as follows:For example,
In particular, this includes as a special case the fundamental formula
This formula distinguishes the complex number i from any real number, since the square of any real number is always a non-negative real number.
With this definition of multiplication and addition, familiar rules for the arithmetic of rational or real numbers continue to hold for complex numbers. More precisely, the distributive property, the commutative properties hold. Therefore, the complex numbers form an algebraic structure known as a field, the same way as the rational or real numbers do.
Complex conjugate, absolute value, argument and division
The complex conjugate of the complex number is defined asIt is also denoted by some authors by. Geometrically, is the "reflection" of about the real axis. Conjugating twice gives the original complex number: A complex number is real if and only if it equals its own conjugate. The unary operation of taking the complex conjugate of a complex number cannot be expressed by applying only the basic operations of addition, subtraction, multiplication and division.
For any complex number , the product
is a non-negative real number. This allows to define the absolute value of z to be the square root
By Pythagoras' theorem, is the distance from the origin to the point representing the complex number z in the complex plane. In particular, the circle of radius one around the origin consists precisely of the numbers z such that. If is a real number, then : its absolute value as a complex number and as a real number are equal.
Using the conjugate, the reciprocal of a nonzero complex number can be computed to be
More generally, the division of an arbitrary complex number by a non-zero complex number equals
This process is sometimes called "rationalization" of the denominator, because it resembles the method to remove roots from simple expressions in a denominator.
The argument of is the angle of the radius with the positive real axis, and is written as, expressed in radians in this article. The angle is defined only up to adding integer multiples of, since a rotation by around the origin leaves all points in the complex plane unchanged. One possible choice to uniquely specify the argument is to require it to be within the interval, which is referred to as the principal value.
The argument can be computed from the rectangular form by means of the arctan function.
Polar form
For any complex number z, with absolute value and argument, the equationholds. This identity is referred to as the polar form of z. It is sometimes abbreviated as.
In electronics, one represents a phasor with amplitude and phase in angle notation:
If two complex numbers are given in polar form, i.e., and, the product and division can be computed as
In other words, the absolute values are multiplied and the arguments are added to yield the polar form of the product. The picture at the right illustrates the multiplication of
Because the real and imaginary part of are equal, the argument of that number is 45 degrees, or . On the other hand, it is also the sum of the angles at the origin of the red and blue triangles are arctan and arctan, respectively. Thus, the formula
holds. As the arctan function can be approximated highly efficiently, formulas like this – known as Machin-like formulas – are used for high-precision approximations of pi|:
Powers and roots
The n-th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product:For example, the first few powers of the imaginary unit i are.
The th roots of a complex number are given by
for. Because sine and cosine are periodic, other integer values of do not give other values. For any, there are, in particular n distinct complex n-th roots. For example, there are 4 fourth roots of 1, namely
In general there is no natural way of distinguishing one particular complex th root of a complex number. One refers to this situation by saying that the th root is a -valued function of.
Fundamental theorem of algebra
The fundamental theorem of algebra, of Carl Friedrich Gauss and Jean le Rond d'Alembert, states that for any complex numbers , the equationhas at least one complex solution z, provided that at least one of the higher coefficients is nonzero. This property does not hold for the field of rational numbers nor the real numbers .
Because of this fact, is called an algebraically closed field. It is a cornerstone of various applications of complex numbers, as is detailed further below.
There are various proofs of this theorem, by either analytic methods such as Liouville's theorem, or topological ones such as the winding number, or a proof combining Galois theory and the fact that any real polynomial of odd degree has at least one real root.
History
The solution in radicals of a general cubic equation, when all three of its roots are real numbers, contains the square roots of negative numbers, a situation that cannot be rectified by factoring aided by the rational root test, if the cubic is irreducible; this is the so-called casus irreducibilis. This conundrum led Italian mathematician Gerolamo Cardano to conceive of complex numbers in around 1545 in his Ars Magna, though his understanding was rudimentary; moreover, he later described complex numbers as being "as subtle as they are useless". Cardano did use imaginary numbers, but described using them as "mental torture". This was prior to the use of the graphical complex plane. Cardano and other Italian mathematicians, notably Scipione del Ferro, in the 1500s created an algorithm for solving cubic equations which generally had one real solution and two solutions containing an imaginary number. Because they ignored the answers with the imaginary numbers, Cardano found them useless.Work on the problem of general polynomials ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher. Complex numbers thus form an algebraically closed field, where any polynomial equation has a root.
Many mathematicians contributed to the development of complex numbers. The rules for addition, subtraction, multiplication, and root extraction of complex numbers were developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for the complex numbers was further developed by the Irish mathematician William Rowan Hamilton, who extended this abstraction to the theory of quaternions.
The earliest fleeting reference to square roots of negative numbers can perhaps be said to occur in the work of the Greek mathematician Hero of Alexandria in the 1st century AD, where in his Stereometrica he considered, apparently in error, the volume of an impossible frustum of a pyramid to arrive at the term in his calculations, which today would simplify to. Negative quantities were not conceived of in Hellenistic mathematics and Hero merely replaced the negative value by its positive
The impetus to study complex numbers as a topic in itself first arose in the 16th century when algebraic solutions for the roots of cubic and quartic polynomials were discovered by Italian mathematicians. It was soon realized that these formulas, even if one were interested only in real solutions, sometimes required the manipulation of square roots of negative numbers. In fact, it was proved later that the use of complex numbers is unavoidable when all three roots are real and distinct. However, the general formula can still be used in this case, with some care to deal with the ambiguity resulting from the existence of three cubic roots for nonzero complex numbers. Rafael Bombelli was the first to address explicitly these seemingly paradoxical solutions of cubic equations and developed the rules for complex arithmetic, trying to resolve these issues.
The term "imaginary" for these quantities was coined by René Descartes in 1637, who was at pains to stress their unreal nature:
A further source of confusion was that the equation seemed to be capriciously inconsistent with the algebraic identity, which is valid for non-negative real numbers and, and which was also used in complex number calculations with one of, positive and the other negative. The incorrect use of this identity in the case when both and are negative, and the related identity, even bedeviled Leonhard Euler. This difficulty eventually led to the convention of using the special symbol in place of to guard against this mistake. Even so, Euler considered it natural to introduce students to complex numbers much earlier than we do today. In his elementary algebra text book, Elements of Algebra, he introduces these numbers almost at once and then uses them in a natural way throughout.
In the 18th century complex numbers gained wider use, as it was noticed that formal manipulation of complex expressions could be used to simplify calculations involving trigonometric functions. For instance, in 1730 Abraham de Moivre noted that the identities relating trigonometric functions of an integer multiple of an angle to powers of trigonometric functions of that angle could be re-expressed by the following de Moivre's formula:
In 1748, Euler went further and obtained Euler's formula of complex analysis:
by formally manipulating complex power series and observed that this formula could be used to reduce any trigonometric identity to much simpler exponential identities.
The idea of a complex number as a point in the complex plane was first described by Danish–Norwegian mathematician Caspar Wessel in 1799, although it had been anticipated as early as 1685 in Wallis's A Treatise of Algebra.
Wessel's memoir appeared in the Proceedings of the Copenhagen Academy but went largely unnoticed. In 1806 Jean-Robert Argand independently issued a pamphlet on complex numbers and provided a rigorous proof of the fundamental theorem of algebra. Carl Friedrich Gauss had earlier published an essentially topological proof of the theorem in 1797 but expressed his doubts at the time about "the true metaphysics of the square root of −1". It was not until 1831 that he overcame these doubts and published his treatise on complex numbers as points in the plane, largely establishing modern notation and terminology:
If one formerly contemplated this subject from a false point of view and therefore found a mysterious darkness, this is in large part attributable to clumsy terminology. Had one not called +1, −1, positive, negative, or imaginary units, but instead, say, direct, inverse, or lateral units, then there could scarcely have been talk of such darkness.
In the beginning of the 19th century, other mathematicians discovered independently the geometrical representation of the complex numbers: Buée, Mourey, Warren, Français and his brother, Bellavitis.
The English mathematician G.H. Hardy remarked that Gauss was the first mathematician to use complex numbers in "a really confident and scientific way" although mathematicians such as Norwegian Niels Henrik Abel and Carl Gustav Jacob Jacobi were necessarily using them routinely before Gauss published his 1831 treatise.
Augustin-Louis Cauchy and Bernhard Riemann together brought the fundamental ideas of [|complex analysis] to a high state of completion, commencing around 1825 in Cauchy's case.
The common terms used in the theory are chiefly due to the founders. Argand called the direction factor, and the modulus; Cauchy called the reduced form and apparently introduced the term argument; Gauss used for, introduced the term complex number for, and called the norm. The expression direction coefficient, often used for, is due to Hankel, and absolute value, for modulus, is due to Weierstrass.
Later classical writers on the general theory include Richard Dedekind, Otto Hölder, Felix Klein, Henri Poincaré, Hermann Schwarz, Karl Weierstrass and many others. Important work in complex multivariate calculus has been started at beginning of the 20th century. Important results have been achieved by Wilhelm Wirtinger in 1927.